Question

Two particles in a box have a total energy \(5 \pi^{2} \hbar^{2} / 2 m L^{2}\) (a) Which states are occupied? (b) Make a sketch of \(P_{S}\left(x_{1}, x_{2}\right)\) versus \(x_{1}\) for points along the line \(x_{2}=x_{1}\) (c) Make a similar sketch of \(P_{A}\left(x_{1}, x_{2}\right)\). (d) Repeat parts (b) and (c) but for points on the line \(x_{2}=L-x_{1}\). (Note: \(\left.\sin [m \pi(L-x) / L]=(-1)^{m+1} \sin \left(m \pi .0^{\prime} L\right) .\right)\)

Short answer

The states that are occupied are (1, 2) and (2, 1). For detailed graph sketches, please refer to a suitable physics or chemistry textbook. The general trend to note is that on the line \(x_{2} =x_{1}\), the symmetric state has higher probability, while on the line \(x_{2}=L-x_{1}\), the antisymmetric state has higher probability.

Step-by-step solution

Identify the States

Given that the total energy is \(5 \pi^{2} \hbar^{2} / 2 m L^{2}\), we should find out the possible states. The formula for the energy of a particle in a quantum box is \(E = \frac{n^{2} \pi^{2} \hbar^{2}}{2mL^{2}}\). Therefore, the possible states will be the pairs of numbers \((n1,n2)\) such that \(n_{1}^2 + n_{2}^2 = 5\)

Solve the Equation for States

The solutions to this equation will give us the possible states. In this case, the pairs would be \((1, 2)\), \((2, 1)\). These are the states that are occupied.

Understand the Probability Distribution Function

The probability distribution function for the symmetric and antisymmetric states are given by \(P_{S}(x_1, x_2) = |\Psi_{S}(x_1, x_2)|^2\) and \(P_{A}(x_1, x_2) = |\Psi_{A}(x_1, x_2)|^2\) respectively, where \(\Psi_{S}(x_1, x_2)\) and \(\Psi_{A}(x_1, x_2)\) are the wave functions of the symmetric and antisymmetric states respectively. These functions can be obtained using the wave functions for the allowed energy states and swapping particles 1 and 2.

Consider the trajectories \(x_{2}=x_{1}\) and \(x_{2}=L-x_{1}\)

For the first line \(x_{2} =x_{1}\) we can obtain the curve for both symmetric and antisymmetric states, and by using the equation provided in the note for the second line \(x_{2}=L-x_{1}\). Due to the limitations of text-based response, we cannot provide a graph here, but it should be noted that on the line \(x_{2} =x_{1}\), the symmetric state will have higher probability, while on the line \(x_{2}=L-x_{1}\), the antisymmetric state will have higher probability.

Key concepts

Particle in a box

In quantum mechanics, the concept of a "particle in a box" serves as a fundamental model. It helps us understand how particles behave when confined to a limited space. Imagine a particle that can't escape the walls of a box, and we're interested in knowing its possible energy states.
The box is essentially a one-dimensional system with infinitely high walls that prevent the particle from leaving. The particle's allowed energy levels arise from the constraints of the box. This is because the particle's wave functions must satisfy boundary conditions at the walls. These conditions allow only certain wave functions, or "stationary states," which correspond to quantized energy levels.
One of the key equations used is:

• The energy of a state: \[ E = \frac{n^{2} \pi^{2} \hbar^{2}}{2mL^{2}} \] where \(n\) is the quantum number, \(m\) is the mass of the particle, and \(L\) is the length of the box.

These quantized energy levels mean the particle can only occupy certain states, determined by quantum numbers \((n1, n2)\).

Energy levels

The energy levels of a particle in a box are quantized due to the constraints imposed by the walls of the box. These energy levels are determined by the quantum numbers \(n\), which can only take on integer values because of the nature of the wave function solutions in bounded conditions.
When we calculate these energy levels using the formula \(E = \frac{n^{2} \pi^{2} \hbar^{2}}{2mL^{2}}\), we learn that each state is associated with a specific amount of energy. In the original exercise, it specifies a total energy, allowing us to deduce the occupied states.
For example, with the total energy given as \(5 \pi^{2} \hbar^{2} / 2 m L^{2}\), we solve \(n_1^2 + n_2^2 = 5\). Here, the possible pairs of states are \((1,2)\) and \((2,1)\). Each of these corresponds to different configurations that the particles can occupy within the box. Understanding these states gives insight into the discrete nature of quantum systems.

Wave functions

Wave functions play a crucial role in quantum mechanics, as they describe the quantum state of a particle. Specifically, for a particle in a box, wave functions define how the probability of finding a particle changes with position and time.
The wave function for a particle confined in a box has a sinusoidal form and satisfies the boundary conditions of the box, making it zero at the walls. For example, the wave function for the \(n\)-th energy level is given by:

• \(\Psi_{n}(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)\) where \(L\) is the length of the box.

Different states of the system correspond to these different wave functions, which oscillate between the walls. By plotting these wave functions, we can observe nodes, points where the wave function is zero. The number of nodes increases with the energy level \(n\).
Symmetric and antisymmetric states are formed for systems with multiple particles, which have slightly different wave functions derived from these single-particle states.

Probability distribution

In quantum mechanics, the probability distribution function tells us where we are most likely to find a particle at any given time. For the case of particles in a box, this is given by the square of the absolute value of the wave function, \(|\Psi(x)|^2\).
This relates directly to the probability density for symmetric and antisymmetric states:

• Symmetric probability distribution: \(P_{S}(x_1, x_2) = |\Psi_{S}(x_1, x_2)|^2\)
• Antisymmetric probability distribution: \(P_{A}(x_1, x_2) = |\Psi_{A}(x_1, x_2)|^2\)

In symmetric states, the probability is more evenly spread, which aligns with intuitive predictions that during identical particle swaps, the system doesn't change. Antisymmetric states, however, require that swapping particles will change the sign of the wave function, leading to a different probability distribution.
In particular lines such as \(x_2 = x_1\) and \(x_2 = L-x_1\), these distributions reveal unique properties about particle interactions and placements, crucial for understanding multi-particle quantum systems.